There are many ways you can do this. I suggest using logistic regression simply because it seems a lot simpler than really complex contingency tables (as I think you agree). You can just specify interaction terms among the design variables created from your categorical variable.

I am confused on what your IV and DV are as in your first sentence you only have dependent variables listed. If your IV are all 2 level categorical variables (like gender) its fairly easy to analyze interaction. If there are more than two levels of your categorical variable than analyzing interaction could be a bit tricky.

But the way you do it is fairly straight forward. You just specify interaction terms between the variables you think interact and then see if they are statistically signficant. So if you have two nutrition types with two levels each (say A and B) than you specify A*B as an interaction effect and test that. If you have three two level nutrition variables (A, B, C) you would have three two way interaction terms (A*B A*C B*C) and one three way interactions A*B*C.

Note that as you get beyond 3 way interaction it gets really difficult to explain what they mean I actually have never seen such an analysis.